None. We are just using what the problem is saying. Say that is such a function so that . This means for any . Thus, has property that for any .
This is just using the definition of what the function is.

None. We are just using what the problem is saying. Say that is such a function so that . This means for any . Thus, has property that for any .
This is just using the definition of what the function is.

Very interesting your function concept,but then again what allow us for the creation of a function.

in the following example how the function concept could be used?

suppose we want to prove that:

...................(a+b+c)/3 >= (abc)^1/3,with a,b,c>=0

Now without using the Am-Gm concept for the proof of the inequality,
one way of solving the inequality is to use the substitution

Very interesting your function concept,but then again what allow us for the creation of a function.

in the following example how the function concept could be used?

suppose we want to prove that:

...................(a+b+c)/3 >= (abc)^1/3,with a,b,c>=0

Now without using the Am-Gm concept for the proof of the inequality,
one way of solving the inequality is to use the substitution

......................x=(a+b+c)/3

I would like to give a nice proof for this inequality:

which is the desired result (also see the proof for Young's inequality).

As ThePerfectHacker mentioned, there is no axiom on the things you told.
You just find your way yourself by picking new parameters or arranging the given ones to reveal your own way.
Ofcourse, experience is the most important thing in this dark forest!

I am sorry, but i forgot to say that in step wise proof ( = formal) where each step is justified ,when a substitution of this type is done,( x=a+b+c or x=a+b or e.t.c e.t.c ) one will need to justified that

PerfectHacker brought in the concept of function and i was forced to point out that even the concept of function needs a theorem of existence.

Are you asking why we know there is such a function such that ? Is that why you mean by existence? If so it does not matter - that was simply for an example. We can assume it exists. And see where we get from there.

Are you asking why we know there is such a function such that ? Is that why you mean by existence? If so it does not matter - that was simply for an example. We can assume it exists. And see where we get from there.

As i said in a step wise proof, that everything is justified we cannot assume anything to exist unless you do not accept formal proofs